Composite object resting on a horizontal plane

[John Cartwell]

A solid hemisphere has another, smaller solid hemisphere attached to each other at their plane, circular faces, so that the centre C of the circular bases of either coincide. A particle P is then attached to the edge of the plane circular face of the larger hemisphere. The composite object is then rested in equilibrium on a horizontal plane, so that it is in contact with the horizontal plane at the point X on the curved surface of the larger hemisphere. Why is X vertically below C?

https://www.physicsforums.com/attachments/imga0303-jpg.74892/

 

[berkeman]

A solid hemisphere has another, smaller solid hemisphere attached to each other at their plane, circular faces, so that the centre C of the circular bases of either coincide. A particle is then attached to the edge of the plane circular face of the larger hemisphere. The composite object is then rested in equilibrium on a horizontal plane, so that it is in contact with the horizontal plane at the point X on the curved surface of the larger hemisphere. Why is X vertically below C?

Welcome to the PF.

What is the context for your question? Is this for schoolwork?

 

[John Cartwell]

No, it is not for schoolwork.

 

[berkeman]

No, it is not for schoolwork.

Can you take a cut at solving for why X is below C?

 

[John Cartwell]

I don't know why X is below C. It could have something to do with moments. Maybe if X is not vertically below C there will be a moment about some point.

 

[John Cartwell]

Could I ask you something? Why have you removed my question?

 

[berkeman]

Could I ask you something? Why have you removed my question?

Removed? I moved it from a Physics forum to a Math forum, since you are asking a geometry question. The fit seemed better. If I thought it was a schoolwork question, I would have moved it to the Pre-Calculus Math Homework forum.

 

[John Cartwell]

Oh, I see. Sorry

 

[Mark44]

A solid hemisphere has another, smaller solid hemisphere attached to each other at their plane, circular faces, so that the centre C of the circular bases of either coincide. A particle is then attached to the edge of the plane circular face of the larger hemisphere. The composite object is then rested in equilibrium on a horizontal plane, so that it is in contact with the horizontal plane at the point X on the curved surface of the larger hemisphere. Why is X vertically below C?

https://www.physicsforums.com/attachments/74891

What does the small hemisphere have to do with anything? Is X the point of contact of the large hemisphere with the plane surface? If so, the center of the hemisphere will always be directly above the point of contact, as long as the plane surface is horizontal.

 

[berkeman]

What does the small hemisphere have to do with anything? Is X the point of contact of the large hemisphere with the plane surface? If so, the center of the hemisphere will always be directly above the point of contact, as long as the plane surface is horizontal.

But is that a stable position? It does seem like the center of mass is off to the side -- would the bottom hemisphere not want to right itself?

 

[John Cartwell]

The centre of mass is balanced by the particle P

 

[Mark44]

But is that a stable position? It does seem like the center of mass is off to the side -- would the bottom hemisphere not want to right itself?

That extra weight at P is offset by the portion of the two hemispheres that are above and to the left of the center point C. Since the whole thing is balanced, the contact point (X) would be directly below the center point (C).

Again, I don't see that the small hemisphere does anything but confuse things.

The way I'm thinking about this is a solid hemisphere sitting on a flat surface. With no extra weight on it, it will be sitting so that its cut edge will be horizontal. If you apply a force down on one side, you lift the opposite part of the hemisphere so that the portion above a horizontal line going through C is exactly balanced by the force downward on the right side. The contact point X would be directly below the center, which will always be the case, due to the spherical shape of the hemisphere.

That's my take at any rate.

 

[John Cartwell]

Mark 44: The smaller hemisphere is not really relevant.

Could you explain why the centre of the hemisphere will always be directly above the point of contact as long as the plane surface is horizontal.

 

[Mark44]

Mark 44: The smaller hemisphere is not really relevant.

Could you explain why the centre of the hemisphere will always be directly above the point of contact as long as the plane surface is horizontal.

Because each point on a sphere is the same distance from the center. It's simpler to think in two dimensions, such as with a circle that rolls along a horizontal line. No matter how the circle is rotated, the point of contact is directly below the circle's center.

BTW, I moved the image to your first post, and deleted the conversation between you and berkeman about posting an image.

 

[John Cartwell]

Still don't see why that explains it.

 

[Mark44]

What part don't you understand?

 

[John Cartwell]

Think I have a glimmering of understanding now. If someone is riding a bicycle, the point of contact will always be directly below the centre of the wheel. Will think about that.

 

[Mark44]

Think I have a glimmering of understanding now. If someone is riding a bicycle, the point of contact will always be directly below the centre of the wheel.

Yes. The contact point can't get ahead of or behind the center of the wheel without deforming the wheel shape.

 

[John Cartwell]

Tangent to a circle is perpendicular to the radius at the point of contact.